Development of Lee’s exact method for Gauss–Krüger projection

Abstract

Lee’s exact method was developed to enable the Gauss–Krüger (GK) projection to be implemented without iterative procedures via the expansion of the intermediate mapping (Thompson projection) into series approximations in terms of isothermal coordinates $\left( \psi , \lambda \right)$ for the forward mapping and GK coordinates $\left( x, y \right)$ for the reverse mapping. The straightforward procedures expressed by new formulas for both forward and reverse mapping of the GK projection were composed by three sequential steps which essentially reveal the mapping procedures and intrinsic properties of the GK projection: The first step of deriving the isothermal coordinates $\left( \psi , \lambda \right)$ from the geodetic coordinates $\left( \varphi , \lambda \right)$ specifies a conformal mapping (i.e., the Normal Mercator projection) of the Earth ellipsoid surface into the Euclidean plane excluding the South and North poles, and the subsequent two steps allow the GK projection to be expressed analytically via the elliptic functions and integrals. Based on the three-step procedure, the conformality and singularities over the entire ellipsoid of the Normal Mercator, Thompson and GK projections were analyzed and the fundamental domains of them were determined. With respect to the precision and efficiency, it was verified that the new algorithm and the complex latitude method had equivalent precision levels for the same orders of the third flatting n with the Krüger-n series from $n^2$ to $n^{12}$ but slower about 0.19 to ${0.21}\,\upmu \hbox {s}$ than the Krüger-n series for a GK coordinates calculating. However, the new formulas provide series approximations for the forward mapping of the Thompson projection and projective transformations for the Normal Mercator, Thompson and GK projections.

Appendix: Lists of series’ coefficients

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} a_1= 1 \\ a_2=\frac{4 n \left( 3+2 n+3 n^2\right) }{3 (1+n)^4} \\ a_3= \frac{64 n^2 \left( 1+n+n^2\right) \left( 5+2 n+5 n^2\right) }{15 (1+n)^8} \\ a_4= \frac{256 n^3 \left( 1+n+n^2\right) }{315 (1+n)^{12}} \left( 147+172 n+322 n^2 +172 n^3+147 n^4\right) \\ a_5=\frac{1024 n^4 \left( 1+n+n^2\right) }{2835 (1+n)^{16}}\left( 1881+3690 n+7511 n^2 +7436 n^3+7511 n^4+3690 n^5+1881 n^6\right) \\ a_6= \frac{4096 n^5 \left( 1+n+n^2\right) }{155925 (1+n)^{20}} \left( 148005+409468 n +954756 n^2+1316484 n^3+1600174 n^4 \right. \\ \qquad \qquad \qquad \qquad \qquad \left. +1316484 n^5+954756 n^6+409468 n^7+148005 n^8\right) \\ \end{array}\right. } \end{aligned} \end{aligned}$$

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$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} {a}'_1= 1 \\ {a}'_2= -\frac{4 n \left( 3+2 n+3 n^2\right) }{3 (1-n)^4} \\ {a}'_3= \frac{16 n^2 \left( 25+32 n+62 n^2+32 n^3+25 n^4 \right) }{15 (1-n)^8} \\ {a}'_4= - \frac{256 n^3 }{315 (1-n)^{12}} \left( 252+473 n+1096 n^2 +1030 n^3+1096 n^4+473 n^5+252 n^6 \right) \\ {a}'_5=\frac{256 n^4 }{2835 (1-n)^{16}}\left( 18621+45936 n+122524 n^2 +160976 n^3+212174 n^4+160976 n^5 \right. \\ \qquad \qquad \qquad \qquad \qquad \left. +122524 n^6+45936 n^7+18621 n^8 \right) \\ {a}'_6= - \frac{1024 n^5 }{155925 (1-n)^{20}} \left( 2185425+6672418 n+20019565 n^2 ~ +33138104 n^3+51507906 n^4 \right. \\ ~~~~~~~~~~~~~~~~~ \left. +53289548 n^5+51507906 n^6 +33138104 n^7+20019565 n^8+6672418 n^9+2185425 n^{10} \right) \\ \end{array}\right. } \end{aligned} \end{aligned}$$

(67)

$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} b_1= 3 n-\frac{4 n^2}{3}-\frac{47 n^3}{24}+\frac{293 n^4}{90}-\frac{241 n^5}{720} -\frac{72029 n^6}{18900}+\frac{7021093 n^7}{1612800}+\frac{5196481 n^8}{25401600}-\cdots \\ b_2= \frac{73 n^2}{12}-\frac{76 n^3}{15}-\frac{3427 n^4}{360}+\frac{6278 n^5}{315} -\frac{81547 n^6}{69120}-\frac{5357593 n^7}{151200}+\frac{348016423 n^8}{8709120} +\cdots \\ b_3= \frac{531 n^3}{40}-\frac{1121 n^4}{70}-\frac{144841 n^5}{4480} \qquad ~+\frac{365357 n^6}{4320}-\frac{3427253 n^7}{537600}-\frac{524932337 n^8}{2661120}+\cdots \\ b_4= \frac{603793 n^4}{20160}-\frac{14843 n^5}{315}-\frac{12470017 n^6}{129600} +\frac{16018067 n^7}{51975}-\frac{38515416901 n^8}{958003200} -\cdots \\ b_5= \frac{555379 n^5}{8064}-\frac{3803159 n^6}{28512} -\frac{568876163 n^7}{2128896}+\frac{64225156139 n^8}{62270208}-\cdots \\ b_6= \frac{4266870481 n^6}{26611200}-\frac{885241627 n^7}{2402400} -\frac{3440458049657 n^8}{4843238400}+\cdots \\ b_7= \frac{37217872147 n^7}{98841600}-\frac{44517464099 n^8}{44478720}- \cdots \\ b_8= \frac{41377942693441 n^8}{46495088640}-\cdots \end{array}\right. } \end{aligned} \end{aligned}$$

(68)

$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} c_1= 1-3 n+\frac{89 n^2}{12}-\frac{983 n^3}{60}+\frac{669097 n^4}{20160}-\frac{85205 n^5}{1344} +\frac{4613096261 n^6}{39916800}-\frac{7023747629 n^7}{34594560}+\frac{241216793515063 n^8}{697426329600}-\cdots \\ c_2= 6 n-\frac{154 n^2}{3}+\frac{8267 n^3}{30}-\frac{146663 n^4}{126}+\frac{14143481 n^5}{3360} -\frac{67912497689 n^6}{4989600} +\frac{3492863985997 n^7}{86486400}-\frac{609515305090687 n^8}{5448643200}+\cdots \\ c_3= \frac{146 n^2}{3}-\frac{10166 n^3}{15}+\frac{1151753 n^4}{210}-\frac{21027229 n^5}{630} +\frac{24032441159 n^6}{142560} -\frac{1786716407701 n^7}{2402400}+\frac{153372726415967 n^8}{51891840}-\cdots \\ c_4= \frac{2124 n^3}{5}-\frac{2576596 n^4}{315}+\frac{9257249 n^5}{105}-\frac{3103882183 n^6}{4455} +\frac{9696749750707 n^7}{2162160}-\frac{96707303981543 n^8}{3891888}+ \cdots \\ c_5= \frac{1207586 n^4}{315}-\frac{9889618 n^5}{105}+\frac{15800798789 n^6}{12474} -\frac{3320773490077 n^7}{270270}+\frac{3730389142464631 n^8}{38918880}- \cdots \\ c_6= \frac{2221516 n^5}{63}-\frac{164256182516 n^6}{155925}+\frac{36404672059 n^7}{2145} -\frac{131726046437189 n^8}{675675}+\cdots \\ c_7= \frac{17067481924 n^6}{51975}-\frac{7804602117964 n^7}{675675} +\frac{1318332827886247 n^8}{6081075}- \cdots \\ c_8=\frac{297742977176 n^7}{96525}-\frac{81759539103832 n^8}{654885}+\cdots \\ c_9= \frac{82755885386882 n^8}{2837835}-\cdots ~~. \end{array}\right. } \end{aligned} \end{aligned}$$

(69)

$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} d_1= 3 n-\frac{89 n^2 }{12}+\frac{983 n^3}{60}-\frac{669097 n^4}{20160}+\frac{85205 n^5}{1344} -\frac{4613096261 n^6}{39916800}+\frac{7023747629 n^7}{34594560}-\frac{241216793515063 n^8}{697426329600}+\cdots \\ d_2= \frac{73 n^2}{6}-\frac{1214 n^3}{15}+\frac{1229327 n^4}{3360}-\frac{857341 n^5}{630} +\frac{847736173 n^6}{190080}-\frac{13721211301 n^7}{1029600}+\frac{123072262642187 n^8}{3321077760}-\cdots \\ d_3= \frac{354 n^3}{5}-\frac{225179 n^4}{280}+\frac{1173191 n^5}{210}-\frac{134307704 n^6}{4455} +\frac{199038212267 n^7}{1441440}-\frac{51797585778073 n^8}{92252160}+\cdots \\ d_4= \frac{603793 n^4}{1260}-\frac{819668 n^5}{105}+\frac{2298035959 n^6}{31185} -\frac{70583980717 n^7}{135135}+\frac{3821533260867121 n^8}{1245404160}- \cdots \\ d_5= \frac{1110758 n^5}{315}-\frac{173808307 n^6}{2310}+\frac{16232813323 n^7}{18018} -\frac{137037324802363 n^8}{17297280}+ \cdots \\ d_6= \frac{4266870481 n^6}{155925}-\frac{32607070924 n^7}{45045} +\frac{454957622901911 n^8}{43243200}- \cdots \\ d_7=\frac{148871488588 n^7}{675675}-\frac{12320936262697 n^8}{1769040}+\cdots \\ d_8= \frac{41377942693441 n^8}{22702680}- \cdots \end{array}\right. } \end{aligned} \end{aligned}$$

(70)

$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} f_1= \frac{3 n}{2}-\frac{2 n^2}{3}-\frac{47 n^3}{48}+\frac{293 n^4}{180}-\frac{241 n^5}{1440} -\frac{72029 n^6}{37800}+\frac{7021093 n^7}{3225600}+\frac{5196481 n^8}{50803200}-\cdots \\ f_2= \frac{73 n^2}{48}-\frac{19 n^3}{15}-\frac{3427 n^4}{1440}+\frac{3139 n^5}{630}-\frac{81547 n^6}{276480} -\frac{5357593 n^7}{604800}+\frac{348016423 n^8}{34836480}+\cdots \\ f_3= \frac{177 n^3}{80}-\frac{1121 n^4}{420}-\frac{144841 n^5}{26880}+\frac{365357 n^6}{25920} -\frac{3427253 n^7}{3225600}-\frac{524932337 n^8}{15966720}+\cdots \\ f_4= \frac{603793 n^4}{161280}-\frac{14843 n^5}{2520}-\frac{12470017 n^6}{1036800} +\frac{16018067 n^7}{415800}-\frac{38515416901 n^8}{7664025600}- \cdots \\ f_5= \frac{555379 n^5}{80640}-\frac{3803159 n^6}{285120}-\frac{568876163 n^7}{21288960} +\frac{64225156139 n^8}{622702080}- \cdots \\ f_6= \frac{4266870481 n^6}{319334400}-\frac{885241627 n^7}{28828800} -\frac{3440458049657 n^8}{58118860800}+\cdots \\ f_7=\frac{37217872147 n^7}{1383782400}-\frac{44517464099 n^8}{622702080}-\cdots \\ f_8= \frac{41377942693441 n^8}{743921418240}-\cdots \end{array}\right. } \end{aligned} \end{aligned}$$

(71)

$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} g_1= n-\frac{29 n^2}{12}+\frac{91 n^3}{20}-\frac{160561 n^4}{20160}+\frac{272689 n^5}{20160} -\frac{891284321 n^6}{39916800}+\frac{18640929389 n^7}{518918400}-\frac{7889377772219 n^8}{139485265920}+\cdots \\ g_2= \frac{13 n^2}{6}-\frac{194 n^3}{15} +\frac{172103 n^4}{3360}-\frac{106159 n^5}{630} +\frac{659041567 n^6}{1330560}-\frac{29059814447 n^7}{21621600}+\frac{56935382676391 n^8}{16605388800}+\cdots \\ g_3= \frac{122 n^3}{15}-\frac{69337 n^4}{840}+\frac{107501 n^5}{210}-\frac{77589203 n^6}{31185} + \frac{134489990773 n^7}{12972960}-\frac{32131705046897 n^8}{830269440}+\cdots \\ g_4= \frac{49561 n^4}{1260}-\frac{181876 n^5}{315}+\frac{153903661 n^6}{31185} -\frac{12942105289 n^7}{405405}+\frac{214851406472257 n^8}{1245404160}+\cdots \\ g_5= \frac{69458 n^5}{315}-\frac{1978379 n^6}{462}+\frac{1409842051 n^7}{30030} - \frac{6583372516843 n^8}{17297280}+\cdots \\ g_6= \frac{212378941 n^6}{155925}-\frac{67083436804 n^7}{2027025} +\frac{11530411221457 n^8}{25945920}+\cdots \\ g_7= \frac{6089027156 n^7}{675675}-\frac{25661491098559 n^8}{97297200}+\cdots \\ g_8=\frac{1424729850961 n^8}{22702680} +\cdots \\ \end{array}\right. } \end{aligned} \end{aligned}$$

(72)

$$\begin{aligned} \left\{ \begin{array}{ll} k_1= n-\frac{3 n^3}{16}+\frac{15 n^5}{256}+\frac{3 n^7}{4096}+\cdots \\ k_2= -\frac{n^2}{4}-\frac{29 n^6}{1024}-\frac{13 n^8}{1024}-\cdots \\ k_3= \frac{n^3}{16}+\frac{n^5}{64}+\frac{31 n^7}{2048}+\cdots \\ k_4= -\frac{n^4}{64}-\frac{n^6}{128}-\frac{7 n^8}{1024}- \cdots \\ k_5= \frac{n^5}{256}+\frac{3 n^7}{1024}+ \cdots \\ k_6= -\frac{n^6}{1024}-\frac{n^8}{1024}-\cdots \\ k_7= \frac{n^7}{4096}+\cdots \\ k_8= -\frac{n^8}{16384}- \cdots \\ \end{array} \right. \qquad \left\{ \begin{array}{ll} l_1= n-\frac{7 n^3}{16}+\frac{71 n^5}{256}-\frac{4201 n^7}{36864}+\cdots \\ l_2= \frac{3 n^2}{4}-\frac{5 n^4}{6}+\frac{2317 n^6}{3072}-\frac{8173 n^8}{15360}+\cdots \\ l_3= \frac{13 n^3}{16}-\frac{95 n^5}{64}+\frac{18743 n^7}{10240}- \cdots \\ l_4= \frac{197 n^4}{192}-\frac{5039 n^6}{1920}+\frac{21233 n^8}{5120}- \cdots \\ l_5= \frac{361 n^5}{256}-\frac{42833 n^7}{9216}+ \cdots \\ l_6= \frac{10471 n^6}{5120}-\frac{295949 n^8}{35840}+ \cdots \\ l_7= \frac{567409 n^7}{184320}-\cdots \\ l_8= \frac{8191879 n^8}{1720320}- \cdots \\ \end{array}\right. \end{aligned}$$

(73)

$$\begin{aligned}&\begin{aligned} {\left\{ \begin{array}{ll} m_1= 4 n-\frac{32 n^2}{3}+\frac{124 n^3}{5}-\frac{3296 n^4}{63}+\frac{32476 n^5}{315}-\frac{30081056 n^6}{155925}+\frac{702328028 n^7}{2027025}-\frac{25612953664 n^8}{42567525}+ \cdots \\ m_2= \frac{56 n^2}{3}-\frac{1984 n^3}{15}+\frac{65872 n^4}{105}-\frac{764096 n^5}{315}+\frac{85344488 n^6}{10395}-\frac{221278592 n^7}{8775}+\frac{29114859808 n^8}{405405}-\cdots \\ m_3= \frac{1792 n^3}{15}-\frac{149984 n^4}{105}+\frac{1085824 n^5}{105}-\frac{1801378688 n^6}{31185}+\frac{110474417024 n^7}{405405}-\frac{460933782976 n^8}{405405}+\cdots \\ m_4= \frac{273856 n^4}{315}-\frac{931328 n^5}{63}+\frac{4510583296 n^6}{31185}-\frac{428596936192 n^7}{405405}+\frac{7759219530112 n^8}{1216215}-\cdots \\ m_5= \frac{2137088 n^5}{315}-\frac{57822208 n^6}{385}+\frac{27867235328 n^7}{15015}-\frac{2267235680768 n^8}{135135}+ \cdots \\ m_6= \frac{1232232448 n^6}{22275}-\frac{3063276371968 n^7}{2027025}+\frac{6541987361792 n^8}{289575}- \cdots \\ m_7= \frac{314093993984 n^7}{675675}-\frac{18390588870656 n^8}{1216215}+\cdots \\ m_8= \frac{11331532414976 n^8}{2837835}-\cdots \end{array}\right. } \end{aligned} \end{aligned}$$

(74)